On Gittins’ index theorem in continuous time

We give a new and comparably short proof of Gittins’ index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.     

On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

We consider the equation \(\rm f^{\prime\prime}+{A}(z){f}=0\) with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of \(\rm f^{\prime\prime}+{e}^P{f}=0\) satisfies λ(f) = ∞. Finally we consider the particular equation \(\rm f^{\prime\prime}+({e}^Z-K){f}=0\) where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.     

Metal derivatives of azoles,part VII. Pyrazolato-bridged dinuclear complexes containing either cyclopalladated or -platinated trimesitylphosphine or -arsine

Summary The halogen bridges of the dimeric, cyclometallated trimesityl-arsine and -phosphine complexes of palladium(II) and platinum(II), where M=Pd or Pt and E=P or As have been replaced with pyrazolate groups to give the corresponding and less symmetric pyrazolato-bridged complexes, where M=Pd or Pt, E=P or As, Pz=pyrazolato anion, and M=Pd, E=As, Pz=3,5-dimethylpyrazolato anion. In the case of the palladium complexes,¹H. n.m.r. clearly indicates the presence of only one isomer which is most likely to have thetrans configuration while the platinum complexes are mixtures of bothcis andtrans forms.Part VI, ref. 3c     

Synthesis and characterization of poly-β-hydroxybutyrate. II. Synthesis of D-poly-β-hydroxybutyrate and the mechanism of ring-opening polymerization of β-butyrolactone

Synthesis of the naturally occurring polyester, D -poly-β-hydroxybutyrate (PHB) was accomplished by using an optically active monomer. Polymerization of D -(+)-β-butyrolactone (β-BL) of 73% optical purity with a catalyst system of Et₃Al–H₂O produced a polymer with a similar optical activity and essentially identical to the natural polymer as isolated from bacterial cells. This paper describes the synthesis and characterization of this optically active polyester along with a suggested mechanism to account for the observed stereospecific polymerization of β-BL with this catalyst system.     

Are dipolar liquids ferroelectric?

VH and HV depolarized hyper-Rayleigh scattering spectra were measured for liquid solutions of dipolar CH3CN in nondipolar C2Cl4 at T=300 K. The VH spectrum contains a strong narrow peak due to a slowly relaxing longitudinal orientation mode. This peak is absent in the HV spectrum, and it disappears from the VH spectrum when the CH3CN concentration is reduced to 8%. This observation is consistent with a ferroelectric phase transition predicted to occur when rho mu0(2)=9epsilon0kT=49 D2 M.     

p53 mutations in basal cell carcinomas arising in routine users of sunscreens

Sun exposure histories were obtained from a series of patients age 35 or younger following diagnosis and removal of a basal cell carcinoma (BCC). The DNA was extracted from tumor biopsy samples derived from BCC of 10 patients who reported that they did not use sunscreens during youth (age 18 or younger) and 10 patients who routinely employed sunscreens during this age period. Exons 5-9 of the p53 gene were then amplified in three fragments from these samples using a nested polymerase chain reaction (PCR) approach and screened for mutations using an RNA heteroduplex assay. All PCR products displaying evidence of a mutation were sequenced. It was found that 6 of the 10 patients who were not routine sunscreen users displayed mutations in these p53 exons. All of the mutations were located at dipyrimidine sites, five of the six were C-->T transitions and one mutation was a tandem double mutation, consistent with a role for solar UVB in BCC formation. In contrast, only one p53 mutation was detected in the group of 10 patients who routinely employed sunscreens during childhood and adolescence. Hence, a significantly (P = 0.029) lower level of p53 mutations was detected in the BCC obtained from sunscreen users compared with tumors derived from nonusers. These findings suggest that the mechanisms involved in the etiology of skin carcinogenesis differ in sunscreen users compared with people who did not routinely employ sunscreens. These data are also indicative of a protective effect associated with sunscreen use against the formation of p53 mutations. It is possible that the patients who were diagnosed with BCC despite their use of sunscreens possessed a genetic susceptibility for skin cancer formation and developed BCC through a p53-independent pathway. Alternatively, solar UVA wavelengths, that were generally not blocked by the suncare products employed by the sunscreen users, may have played a significant role in BCC development through induction of a mutation(s) in an oncogene and/or a tumor suppressor gene, other than p53, for these patients.